If a student knows the concept of bijection, then it would be easier to pass to him the concept of identifying a set with another. That kind of "identification" concerning me is, for example, "We identify $A$ with $B$ if $A$ is homeomorphic to $B$.". But what is a wise way to achieve this intuitively? For instance, some authors intuitively explain the concept of continuity to a layman or a beginning mathematician by saying something like "A function is continuous if you can draw its graph in one stroke.". This example exemplifies what I mean by a wise way; it magnifies the main idea without losing too much precision. Another nice instance would be that of the concept of manifold; some authors introduce the idea of manifold to a general reader by saying something like "A manifold is a set that locally looks like a Euclidean space.". This "definition" is again intuitive and not too sloppy.
So what is a wise way to intuitively explain the concept of identifying one set with another?
Explain to your (interested?!) friend that they already know what it means to identify two finite collections of objects - they have the same number of elements. You can match up the objects 1:1 in any manner you prefer.
Now examining 'primal' sets is fine, but what if there are relations and structures imposed on them? Then, to be identify the collections you have to preserve all those structures and relations as you match up the objects.
Example: Under a suitably defined male/female structure, there are two homeomorphism between
$\{ John, Tamara, Sally \} $
and
$\{ Patricia, Tom, Laura \} $
John $\mapsto$ Tom
Tamara $\mapsto$ Laura
Sally $\mapsto$ Patricia
AND
John $\mapsto$ Tom
Tamara $\mapsto$ Patricia
Sally $\mapsto$ Laura